Journal of the Korean Data and Information Science Society

The Korean Data and Information Science Society (한국데이터정보과학회)
권호 표지
DOI QR코드

DOI QR Code

A numerical study on option pricing based on GARCH models with normal mixture errors

정규혼합모형의 오차를 갖는 GARCH 모형을 이용한 옵션가격결정에 대한 실증연구

  • Jeong, Seung Hwan (Department of Statistics, Hankuk University of Foreign Studies) ;
  • Lee, Tae Wook (Department of Statistics, Hankuk University of Foreign Studies)
  • 정승환 (한국외국어대학교 통계학과) ;
  • 이태욱 (한국외국어대학교 통계학과)
  • Received : 2017.01.05
  • Accepted : 2017.03.06
  • Published : 2017.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

The option pricing of Black와 Scholes (1973) and Merton (1973) has been widely reported to fail to reflect the time varying volatility of financial time series in many real applications. For example, Duan (1995) proposed GARCH option pricing method through Monte Carlo simulation. However, financial time series is known to follow a fat-tailed and leptokurtic probability distribution, which is not explained by Duan (1995). In this paper, in order to overcome such defects, we proposed the option pricing method based on GARCH models with normal mixture errors. According to the analysis of KOSPI200 option price data, the option pricing based on GARCH models with normal mixture errors outperformed the option pricing based on GARCH models with normal errors in the unstable period with high volatility.

Black와 Scholes (1973)와 Merton (1973)의 옵션 가격결정이론에 대한 논문이 발표 된 이후 다양한 실증 분석 결과에 의하여 시간의 흐름에 따라 변동성이 불변한다고 가정하는 Black-Scholes 모형이 시장의 옵션 가격을 적절히 설명하지 못하고 있다는 것이 밝혀지면서 많은 대안적인 연구들이 진행되어 왔다. 예를 들어, Duan (1995)은 위험중립측도 하에서의 몬테카를로 시뮬레이션을 통해 GARCH 모형을 따르는 기초 자산의 옵션가격을 도출하는 방법을 제시하였다. 그러나 실제 주식이나 환율 등의 금융자료에 수익률분포는 정규분포에 비해 꼬리가 두껍고, 급첨의 형태를 보이는 데 Duan (1995)의 옵션가격 결정 방법은 이를 적절히 반영하지 못하고 있다. 이를 해결하기 위해 본 논문에서는 정규혼합모형의 오차를 갖는 GARCH 모형을 이용한 옵션가격 결정 방법을 제안하고자 한다. KOSPI200 옵션가격 자료를 이용하여 본 논문에서 제시된 옵션가격과 정규분포를 가정한 GARCH 모형에 의해 결정된 옵션가격과 비교한 결과, 금융 자료의 급첨의 성질이 뚜렷한 불안정한 시기인 경우에 오차가 정규혼합모형이라고 가정한 GARCH 모형에 의한 옵션가격 결정의 성과가 월등히 좋아지는 것을 확인할 수 있었다.

Keywords

References

  1. Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637-659 https://doi.org/10.1086/260062
  2. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307-327 https://doi.org/10.1016/0304-4076(86)90063-1
  3. Christoffersen, P., Elkamhi, R., Feunou, B. and Jacobs, K. (2010). Option valuation with conditional heteroskedasticity and non-normality. Review of Financial Studies, 23, 2139-2183. https://doi.org/10.1093/rfs/hhp078
  4. Duan, J. C. (1995). The GARCH option pricing model, Mathematical Finance, 5, 13-32. https://doi.org/10.1111/j.1467-9965.1995.tb00099.x
  5. Engle, R. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of UK Inflation. Econometrica, 50, 987-1008. https://doi.org/10.2307/1912773
  6. Kim, M. and Seo, B. (2016). Constrained parameter estimation in normal mixtures using extension of the EM algorithm, Master Thesis, Sungkyunkwan University, Seoul.
  7. Ko, K. and Son, Y. (2015). Optimal portfolio and VaR of KOSPI200 using one-factor model. Journal of the Korean Data & Information Science Society, 26, 323-334. https://doi.org/10.7465/jkdi.2015.26.2.323
  8. Lee, J. and Song, S. (2016). Comparison of methods of approximating option prices with variance gamma processes. The Korean Journal of Applied Statistics, 29, 181-192. https://doi.org/10.5351/KJAS.2016.29.1.181
  9. Lee, T. (2009). Numerical Study on Jarque-Bera normality test for innovations of ARMA-GARCH models. Journal of the Korean Data & Information Science Society, 20, 453-458.
  10. Lee, T. and Ha. J. (2007). Testing for domestic financial data for the normality of the innovation based on the GARCH(1,1) model. Journal of the Korean Data & Information Science Society, 18, 809-815.
  11. Lee, W. and Chun. H. (2016). A deep learning analysis of the Chinese Yuan's volatility in the onshore and offshore markets. Journal of the Korean Data & Information Science Society, 27, 327-335. https://doi.org/10.7465/jkdi.2016.27.2.327
  12. McLachlan, G. J. and Peel, D. (2000). Finite mixture models, Wiley, New York.
  13. Park, S. and Baek. C. (2014). On multivariate GARCH model selection based on risk management. Journal of the Korean Data & Information Science Society, 25, 1333-1343. https://doi.org/10.7465/jkdi.2014.25.6.1333
  14. Rombouts, J. and Stentoft, L. (2015). Option pricing with asymmetric heteroskedastic normal mixture models. International Journal of Forecasting, 31, 635-650. https://doi.org/10.1016/j.ijforecast.2014.09.002

Cited by

  1. Some limiting properties for GARCH(p, q)-X processes vol.28, pp.3, 2017, https://doi.org/10.7465/jkdi.2017.28.3.697